The Inverse Function Theorem for Smooth Functions

In the setting of Functions of Class Cᵏ: Notation, let $f:U\to\mathbb{R}^{n}$ be smooth on $U$, and let $a\in U$ be a point at which the Jacobian matrix $Df(a)$ is invertible.

(i) Smooth local inverse. There is an open set $W\subseteq\mathbb{R}^{n}$ with $a\in W\subseteq U$ such that $V=f(W)$ is open and $f|_{W}$ is a smooth diffeomorphism of $W$ onto $V$; its inverse function $g:V\to W$ satisfies

$$Dg\bigl(f(x)\bigr)=Df(x)^{-1}\qquad\text{for every }x\in W .$$

In particular, $f$ is a local smooth diffeomorphism at $a$.